A081146 -id:A081146 - cp's OEIS frontend

A081145 a(1)=1; thereafter, a(n) is the least positive integer which has not already occurred and is such that |a(n)-a(n-1)| is different from any |a(k)-a(k-1)| which has already occurred.

1, 2, 4, 7, 3, 8, 14, 5, 12, 20, 6, 16, 27, 9, 21, 34, 10, 25, 41, 11, 28, 47, 13, 33, 54, 15, 37, 60, 17, 42, 68, 18, 45, 73, 19, 48, 79, 22, 55, 23, 58, 94, 24, 61, 99, 26, 66, 107, 29, 71, 115, 30, 75, 121, 31, 78, 126, 32, 81, 132, 35, 87, 140, 36, 91, 147, 38, 96, 155, 39

Offset: 1

Don Reble, Mar 08 2003

nonn

The sequence is a permutation of the positive integers. The inverse is A081146.
Similar to A100707, except that when we subtract we use the largest possible k.
The 1977 paper of Slater and Velez proves that this sequence is a permutation of positive integers and conjectures that its absolute difference sequence (see A308007) is also a permutation. If we call this the "Slater-Velez permutation of the first kind", then they also constructed another permutation (the 2nd kind), for which they are able to prove that both the sequence (A129198) and its absolute difference (A129199) are true permutations. - Ferenc Adorjan, Apr 03 2007
The points appear to lie on three straight lines of slopes roughly 0.56, 1.40, 2.24 (click "graph", or see the Wilks link). I checked this for the first 10^6 terms using Allan Wilks's C program. See A308009-A308015 for further information about the three lines. - N. J. A. Sloane, May 14 2019

			a(4)=7 because the previous term is 4 and the differences |3-4|, |5-4| and |6-4| have already occurred.
After 7 we get 3 as the difference 4 has not occurred earlier. 5 follows 14 as the difference 9 has not occurred earlier.

Ferenc Adorjan, Table of n,a(n) for n = 1..5000
Shalosh B. Ekhad and Doron Zeilberger, Guessing the Elusive Patterns in the Slater-Valez sequence (aka OEIS A081145), June 2019; Local copy
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific Journal of Mathematics, Vol. 71, No. 1, 1977, 193-196.
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, II, Pacific Journal of Mathematics, Vol. 82, No. 2, 1979, 527-531.
William Y. Velez, Research problems 159-160, Discrete Math., 110 (1992), pp. 301-302.
Allan Wilks, Table showing n, a(n), slope, line_number, for n=1..100000 [The three lines are labeled 0 (lower), 1 (middle), 2 (upper).]
Allan Wilks, C program for A081145
Index entries for sequences that are permutations of the natural numbers

The sequence of differences is A099004 (see also A308007).
Similar to Murthy's sequence A093903, Cald's sequence (A006509) and Recamán's sequence A005132. See also A100707 (another version).
A308021 is an offspring of this sequence. - N. J. A. Sloane, May 13 2019
Cf. A063733, A072007, A078783, A081146, A084331, A084335, A117622.
See A308009-A308015 for the lines that the points lie on.
A308172 gives smallest missing numbers.

import Data.List (delete)
a081145 n = a081145_list !! (n-1)
a081145_list = 1 : f 1 [2..] [] where
   f x vs ws = g vs where
     g (y:ys) = if z `elem` ws then g ys else y : f y (delete y vs) (z:ws)
                where z = abs (x - y)
-- Reinhard Zumkeller, Jul 02 2015

f[s_] := Block[{d = Abs[Rest@s - Most@s], k = 1}, While[ MemberQ[d, Abs[k - Last@s]] || MemberQ[s, k], k++ ]; Append[s, k]]; NestList[s, {1}, 70] (* Robert G. Wilson v, Jun 09 2006 *)
f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jun 13 2006 *)

{SV_p1(n)=local(x,v=6,d=2,j,k); /* Slater-Velez permutation - the first kind (by F. Adorjan)*/ x=vector(n);x[1]=1;x[2]=2; for(i=3,n,j=3;k=1;while(k,if(k=bittest(v,j)||bittest(d,abs(j-x[i-1])),j++,v+=2^j;d+=2^abs(j-x[i-1]);x[i]=j))); return(x)} \\ Ferenc Adorjan, Apr 03 2007

A081145_list, l, s, b1, b2 = [1,2], 2, 3, set(), set([1])
for n in range(3, 10**2):
    i = s
    while True:
        m = abs(i-l)
        if not (i in b1 or m in b2):
            A081145_list.append(i)
            b1.add(i)
            b2.add(m)
            l = i
            while s in b1:
                b1.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 15 2014

A099004 First differences of A081145.

Original entry on oeis.org

1, 2, 3, -4, 5, 6, -9, 7, 8, -14, 10, 11, -18, 12, 13, -24, 15, 16, -30, 17, 19, -34, 20, 21, -39, 22, 23, -43, 25, 26, -50, 27, 28, -54, 29, 31, -57, 33, -32, 35, 36, -70, 37, 38, -73, 40, 41, -78, 42, 44, -85, 45, 46, -90, 47, 48, -94, 49, 51, -97, 52, 53, -104, 55, 56, -109, 58, 59, -116, 61, -60, 62, 63

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

Does every number appear in the sequence of absolute values (see A308007)?

Chai Wah Wu, Table of n, a(n) for n = 1..5000

Cf. A081145, A081146, A308007, A308021.

f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; t = Nest[f, {1}, 75]; Rest@t - Most@t (* Robert G. Wilson v, Jun 13 2006 *)

A099004_list, l, s, b1, b2 = [1], 2, 3, set(), set([1])
for n in range(2, 5*10**3+1):
    i = s
    while True:
        m = abs(i-l)
        if not (i in b1 or m in b2):
            A099004_list.append(i-l)
            b1.add(i)
            b2.add(m)
            l = i
            while s in b1:
                b1.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 15 2014

Edited by N. J. A. Sloane, Apr 08 2006

More terms from Robert G. Wilson v, Jun 13 2006

A308007 Absolute values of first differences of A081145.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 7, 8, 14, 10, 11, 18, 12, 13, 24, 15, 16, 30, 17, 19, 34, 20, 21, 39, 22, 23, 43, 25, 26, 50, 27, 28, 54, 29, 31, 57, 33, 32, 35, 36, 70, 37, 38, 73, 40, 41, 78, 42, 44, 85, 45, 46, 90, 47, 48, 94, 49, 51, 97, 52, 53, 104, 55, 56, 109, 58, 59, 116, 61, 60, 62, 63

Offset: 1

N. J. A. Sloane, May 13 2019

nonn

The (signed) differences themselves are in A099004, but this sequence is important enough to have its own entry.
Conjectured (see the Slater-Velez and Velez articles) to be a permutation of the positive integers.
It appears that the terms line on two lines (this is true for the first million terms): see A308016-A308020.

Chai Wah Wu, Table of n, a(n) for n = 1..5000
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific Journal of Mathematics, Vol. 71, No. 1, 1977, 193-196.
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, II, Pacific Journal of Mathematics, Vol. 82, No. 2, 1979, 527-531.
William Y. Velez, Research problems 159-160, Discrete Math., 110 (1992), pp. 301-302.

Cf. A081145, A081146, A099004, A308021.

A308021 Start at n=1. Fill in a(n) with a value d > 0 not used earlier such that n-d or n+d is the smallest possible index not visited earlier, then continue with that index.

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 4, 6, 12, 15, 17, 8, 20, 9, 22, 11, 25, 27, 29, 14, 13, 33, 35, 37, 16, 40, 18, 19, 42, 45, 47, 49, 21, 24, 52, 55, 23, 58, 61, 62, 30, 26, 65, 66, 28, 68, 34, 31, 71, 74, 76, 79, 81, 39, 32, 83, 86, 36, 89, 43, 38, 91, 93, 96, 99, 41, 101, 50, 103, 106, 44, 108, 54, 111, 46, 113, 117, 48, 57, 118, 51, 120, 123

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 09 2019

nonn

A variant of Recamán's sequence: start at n=1, a(1)=1, then iterate: let n' =  n - a(n) if this n' > 0 and was not visited earlier, otherwise n' = n + a(n). Then let n'' <> n' be the smallest index not visited earlier (a(n'') not yet defined) such that the value |n'-n''| was not yet used (meaning not yet a value of any a(i)). Set a(n') = |n'-n''| and continue with n = n'. [Name and Comment suggested by M. F. Hasler at the request of the authors.]
Conjectured to be a permutation of the positive integers. The conjectured inverse permutation is given in A308049. - M. F. Hasler, May 10 2019
From Rémy Sigrist and N. J. A. Sloane, May 13 2019: (Start)
The sequence is given by the following formula. Let R(t) = A081145(t). Then for all t >= 1, a(R(t)) = |R(t+1)-R(t)|.
For example, for t=10, R(10)=20, R(11)=6, and a(R(10)) = a(20) = |6-20| = 14.
Since it is known that {R(t): t>=1} is a permutation of the positive integers (it is the "Slater-Velez permutation of the first kind"), this specifies a(n) for all n.
The connection with the definition as interpreted above by M. F. Hasler is that at step t of the procedure, n' is R(t) = A081145(t), n'' is R(t+1) = A081145(t+1), and we calculate a(R(t)) = |n'-n''| = |R(t)-R(t+1)|.
The conjecture that {a(n)} is a permutation of the positive integers is equivalent to Slater and Velez's conjecture (see references) that the absolute values of the first differences of A081145 are also a permutation of the positive integers. This problem appears to be still unsolved. (End)

			a(1) = 1 drives us to the empty cell a(2) since we can't go further to the left. We fill this cell with the number 2 which is the smallest integer not used before and thus allows us to go to the leftmost possible empty cell, 2 + 2 = 4. (There are no empty places to the left and we can't go to 3 = 2 + 1 since a step 1 has already been used.) So we have a(2) = 2.
a(2) = 2 drives us to the empty cell a(4). We see that the leftmost empty cell a(3) cannot be reached from a(4) since a step of 1 has already been used. We thus fill the cell a(4) with the smallest integer not used before, a(4) = 3.
a(4) = 3 drives us to the empty cell a(7). We see that the leftmost empty cell a(3) can now be reached from a(7) if we fill a(7) with 4; we have thus a(7) = 4.
a(7) = 4 drives us to the empty cell a(3), which is the one we wanted to fill. We fill a(3) with 5 which is the smallest integer not leading to a contradiction, whence a(3) = 5.
a(3) = 5 drives us to the empty cell a(8). We would like to fill this cell with 3, as this 3 would allow us to fill the leftmost empty cell of the sequence - but 3 has been used before; thus we'll have a(8) = 6.
a(8) = 6 drives us to the empty cell a(14). We fill a(14) with 9 as this will allow us to reach the leftmost empty cell of the sequence, whence a(14) = 9.
a(14) = 9 drives us to the empty cell a(5). We fill a(5) with 7 as this is the smallest integer not leading to a contradiction, so we have a(5) = 7, etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..5564
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific Journal of Mathematics, Vol. 71, No. 1, 1977, 193-196.
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, II, Pacific Journal of Mathematics, Vol. 82, No. 2, 1979, 527-531.
William Y. Velez, Research problems 159-160, Discrete Math., 110 (1992), pp. 301-302.

Cf. A005132 (Recamán's sequence), A171884 (injective variant).
Cf. A308049 (conjectured inverse permutation).
See also A081145, A081146, A099004.

{A=vector(N=199); n=1; while (n<=N,S=Set(A); Z=select(t->!t,A,1); for (i=1,#Z,Z[i]!=n||next; setsearch(S, abs(n-z=Z[i]))&& next; A[n]=abs(n-z); n=z; next(2)); break); if(#Z, A[1..Z[1]-!A[Z[1]]], A)} \\ M. F. Hasler, May 09 2019

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A081145 a(1)=1; thereafter, a(n) is the least positive integer which has not already occurred and is such that |a(n)-a(n-1)| is different from any |a(k)-a(k-1)| which has already occurred.

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A099004 First differences of A081145.

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A308007 Absolute values of first differences of A081145.

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A308021 Start at n=1. Fill in a(n) with a value d > 0 not used earlier such that n-d or n+d is the smallest possible index not visited earlier, then continue with that index.

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A378822 Inverse permutation to A378821.

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